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Quantum Reference Frames in Quantum Circuits: Perspective Dependent Entangling Cost and Coherence Entanglement Trade Offs

Salman Sajad Wani, Saif Al-Kuwari

TL;DR

The paper operationalizes quantum reference frames (QRFs) in circuit models by deriving a gate-transform theorem for finite groups, showing that frame changes convert noncovariant local gates into entangling operations while symmetry-commuting gates remain local. For Abelian groups, gates fall into three classes—frame-robust, phase-only, and entangling—leading to a relational circuit complexity that depends on the observer. It demonstrates a lossless coherence–entanglement interconversion across internal frames in a three-qubit $\mathbb{Z}_2$ system, and validates the framework with a three-qubit QRF circuit implemented on IBM Quantum hardware, observing resource redistribution under realistic noise. The work connects QRF theory to practical circuit design, enables frame-aware resource accounting, and points to extensions to non-Abelian/continuous groups and potential metrological advantages from relational entanglement.

Abstract

The perspective-neutral formulation of quantum reference frames (QRFs) treats observers as quantum systems and describes physics relationally from within the composite system. While frame-change maps and frame-invariant resource sums are theoretically understood, their impact on circuit-based quantum information processing has largely remained unexplored. We formulate QRF transformations as circuit compilation rules and, for systems with finite Abelian symmetry described by the regular representation, derive a gate-level dictionary that maps local operations in one frame to their images in another. This yields a group-theoretic classification of gates where symmetry-commuting operators remain local, up to frame-dependent phases, while generic gates are promoted to controlled entangling operations in which the original frame acts as a control register. The resulting frame-dependence entangling-gate count defines a relational circuit complexity where the cost of a computation depends on the internal reference frame of the observer. We instantiate the framework in a three-qubit model and show that the QRF unitary acts as a lossless converter between a purity-based local coherence and concurrence, preserving their invariant sum and giving a concrete realization of the relativity of entanglement. We implement the corresponding circuits on an IBM Quantum superconducting platform, using full state tomography to reconstruct the redistribution of resources between internal frames. The hardware data reproduce the predicted conversion of local coherence into entanglement, and the observed deviations from exact conservation quantify the effect of realistic device noise on relational quantum protocols.

Quantum Reference Frames in Quantum Circuits: Perspective Dependent Entangling Cost and Coherence Entanglement Trade Offs

TL;DR

The paper operationalizes quantum reference frames (QRFs) in circuit models by deriving a gate-transform theorem for finite groups, showing that frame changes convert noncovariant local gates into entangling operations while symmetry-commuting gates remain local. For Abelian groups, gates fall into three classes—frame-robust, phase-only, and entangling—leading to a relational circuit complexity that depends on the observer. It demonstrates a lossless coherence–entanglement interconversion across internal frames in a three-qubit system, and validates the framework with a three-qubit QRF circuit implemented on IBM Quantum hardware, observing resource redistribution under realistic noise. The work connects QRF theory to practical circuit design, enables frame-aware resource accounting, and points to extensions to non-Abelian/continuous groups and potential metrological advantages from relational entanglement.

Abstract

The perspective-neutral formulation of quantum reference frames (QRFs) treats observers as quantum systems and describes physics relationally from within the composite system. While frame-change maps and frame-invariant resource sums are theoretically understood, their impact on circuit-based quantum information processing has largely remained unexplored. We formulate QRF transformations as circuit compilation rules and, for systems with finite Abelian symmetry described by the regular representation, derive a gate-level dictionary that maps local operations in one frame to their images in another. This yields a group-theoretic classification of gates where symmetry-commuting operators remain local, up to frame-dependent phases, while generic gates are promoted to controlled entangling operations in which the original frame acts as a control register. The resulting frame-dependence entangling-gate count defines a relational circuit complexity where the cost of a computation depends on the internal reference frame of the observer. We instantiate the framework in a three-qubit model and show that the QRF unitary acts as a lossless converter between a purity-based local coherence and concurrence, preserving their invariant sum and giving a concrete realization of the relativity of entanglement. We implement the corresponding circuits on an IBM Quantum superconducting platform, using full state tomography to reconstruct the redistribution of resources between internal frames. The hardware data reproduce the predicted conversion of local coherence into entanglement, and the observed deviations from exact conservation quantify the effect of realistic device noise on relational quantum protocols.

Paper Structure

This paper contains 32 sections, 9 theorems, 105 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Finite-Group Quantum Reference Frames
  3. Gate-level quantum reference frames for finite Abelian groups
  4. Model definition and relational perspectives
  5. Resource conservation and trade-offs
  6. Quantum circuits in different reference frames
  7. Relational compilation in the Z2 model
  8. Relational circuit complexity
  9. Experimental validation
  10. Experimental setup
  11. Experimental results
  12. Conclusion
  13. Finite-group QRF map and gate-transform theorem
  14. Setting and notation
  15. Finite-group QRF unitary
  16. ...and 17 more sections

Key Result

Theorem 3.1

Let $G$ be a finite group acting via the right-regular representation $U_R$. Under the change of reference frame $0\to i$ defined by Eq. eq:finite-group-QRF-unitary, the image of a local unitary $\hat{U}_S$ is

Figures (1)

  • Figure 1: Relational circuit compilation for the $\mathbb{Z}_2$ model. The same physical process (Bell preparation) is decomposed into different gate sequences depending on the observer's reference frame. The local resources in frame $C$ (a) are recompiled into entangling resources involving the frame register in frames $B$ (b) and $A$ (c).

Theorems & Definitions (12)

  • Theorem 3.1: Finite-group gate-transform
  • Corollary 3.2: Classification of Abelian gate transformations
  • Proposition A.1: Finite-group QRF unitary
  • proof
  • Proposition A.2: Local operator transform
  • Theorem A.1: Gate-transform theorem for finite groups
  • proof
  • Corollary A.2: Symmetry-commuting gates remain local
  • Corollary A.3: One-dimensional covariance sector
  • Corollary A.4: Generic gates become entangling
  • ...and 2 more